Ptolemy's theorem states the relationship between the diagonals and the sides of a cyclic quadrilateral. It is a powerful tool to apply to problems about inscribed quadrilaterals.
If a quadrilateral is inscribable in a circle, then the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of the opposite sides:
Let's prove this theorem.
Let be a random quadrilateral inscribed in a circle.
The proposition will be proved if
It's easy to see in the inscribed angles that and
Let be a point on such that then since and
Note that Then since and
Therefore, from and we have
We can prove the Pythagorean theorem using Ptolemy's theorem:
Prove that in any right-angled triangle where
Let be a random rectangle inscribed in a circle.
Applying Ptolemy's theorem in the rectangle, we get
But since is a rectangle. Therefore,
Once upon a time, Ptolemy let his pupil draw an equilateral triangle inscribed in a circle before the great mathematician depicted point and joined the red lines with other vertices, as shown below.
Ptolemy: Dost thou see that all the red lines have the lengths in whole integers?
Pupil: Indeed, master! Such an extraordinary point!
Ptolemy: Now if the equilateral triangle has a side length of 13, what is the sum of the three red lengths combined?
In a quadrilateral, if the product of its diagonals is equal to the sum of the products of the pairs of the opposite sides, then the quadrilateral is inscribable.
Given a quadrilateral ,
where equality occurs if and only if is inscribable.
Let be a point inside quadrilateral such that and .
Triangle is similar to triangle , so and .
This gives us another pair of similar triangles: and .
Adding the two equations together,
The equality occurs when lies on , which means is inscribable.
Another proof requires a basic understanding of properties of inversions, especially those relevant to distance.
Consider a circle of radius 1 centred at . Let and be the resultant of inverting points and about this circle, respectively. Using the distance properties of inversion, we have
Thus, Ptolemy's inequality becomes
which is true by triangle inequality. Therefore, Ptolemy's inequality is true. Thus proven.